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238                               CHEMICAL ENGINEERING
Theory of Screening.—For power-driven screens the question of the degree of reedom with which grains of various size pass through the apertures is one that an best be understood from the mathematics of probability and chance.    In the ensuing theory it is assumed that the fragments or grains approach the openings in a layer one grain deep and that there is no interference by other grains with a grain approaching an aperture.    To put the problem in the simplest form let it be assumed, Fig. 13, that there is a square meshed opening of side L   Approaching the border of the square from any direction, and with a rate of speed some elements of which will be indicated later, is a spherical grain of diameter l/n, n being assumed to be any value equal to or greater than one.   Then evidently for the grain to fall entirely within the square it must pursue some path marked by its center projected, such that at some point in its path the center projected will fall inside the concentric square l-l/n.
A little reflection will show that the chances of the grain falling through the square are in the ratio of the area of the inner square to the area between the inner and outer or
72_?i2-^      -2     J.
n "r n2            n ~r n2      n2 - 2n -f 1
2P _ jP             2 _ T           2n - 1
n       n*            n     n2
The inverse ratio fixes the chance of the grain not passing through, and is also a measure of the number of squares the average grain will have to pass over before falling through.1
It will readily be noted that this statement of theory places the problem in its simplest form. It would be expected that if the grain were placed in any position such that its center of gravity were within the square it would fall through the square. All such questions which by extension include impact on the sides of the aperture, the effect of which will depend on many factors, such as angle of impact, size of particle, velocity of its motion, etc., are not considered, nor, as already stated, the question of the interference of the grains with one another. Where there is screening en masse in a layer more than one grain deep, other factors being equal it must be evident efficiency of the screening must be low since the layers nearest to the screen must be disposed of before the \ipper ones can have trials at the apertures.2 Where tonnage is the consideration rather than precision of work mass screening has its efficiency increased bv the interstitial settlement of the fines through the bed of material on a shaking screen. This factor, helpful in giving more efficiency to the
1 Some criticism has been made of using this ratio to express the chance. It has been said that the ratio of the inner square to the outer expresses the chance. What this last ratio gives is a measure of the percentage chance of a grain going through an aperture. Later in the discussion this ratio will be uaed to estimate the number of grains of a given size, out of any selected arbitrary number, which will fall through an aperture of given size. This is a slight extension of the conception under which the theory has been stated. The ratio of the inner area to the area between the inner and outer squares expresses the chance as understood in common parlance. In mundane affairs the limits of chance are infinity or